For a real-valued random variable, $X$, the first moment method, is simply

$P(X\ge\mathbb{E}[X])>0$

This can be extended to the second moment quite easily:

$P(X\ge\mathbb{E}[X]+\sqrt{Var[X]})>0$

The question must be asked: How does one generalize this to higher (probably centralized) moments?